In Mass Effect, is there an actual Earth and all of the other planets in our solar system in game?

Yes. They appear in the Sol system, in the Local Cluster, which is unlocked either around level 20 or after completing the first 2 story missions (not sure which). None of the planets can be landed on, and none can be surveyed (the idea, presumably, being that humans would have consumed most of the valuable resources before necessitating an expansion beyond our own system), but as with all the planets on the Galaxy Map, you can select them and read a little about how they fit into the Mass Effect universe. Earth's moon, given it's latin name of "Luna" in the game, CAN be landed on. There is a side quest that takes place there, acquired via a call from Admiral Hackett, completion of which earns you a choice of specialisation classes.